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Properties

Label 2-1800-5.3-c2-0-26
Degree $2$
Conductor $1800$
Sign $0.945 - 0.326i$
Analytic cond. $49.0464$
Root an. cond. $7.00331$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.22 − 3.22i)7-s + 6.89·11-s + (18.1 + 18.1i)13-s + (0.449 − 0.449i)17-s − 9.89i·19-s + (−10.6 − 10.6i)23-s + 36.2i·29-s + 25.6·31-s + (13.3 − 13.3i)37-s + 3.10·41-s + (2.72 + 2.72i)43-s + (−37.1 + 37.1i)47-s + 28.2i·49-s + (−65.1 − 65.1i)53-s + 80.3i·59-s + ⋯
L(s)  = 1  + (0.460 − 0.460i)7-s + 0.627·11-s + (1.39 + 1.39i)13-s + (0.0264 − 0.0264i)17-s − 0.520i·19-s + (−0.463 − 0.463i)23-s + 1.25i·29-s + 0.828·31-s + (0.359 − 0.359i)37-s + 0.0756·41-s + (0.0634 + 0.0634i)43-s + (−0.790 + 0.790i)47-s + 0.575i·49-s + (−1.23 − 1.23i)53-s + 1.36i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.945 - 0.326i$
Analytic conductor: \(49.0464\)
Root analytic conductor: \(7.00331\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1),\ 0.945 - 0.326i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.424156263\)
\(L(\frac12)\) \(\approx\) \(2.424156263\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-3.22 + 3.22i)T - 49iT^{2} \)
11 \( 1 - 6.89T + 121T^{2} \)
13 \( 1 + (-18.1 - 18.1i)T + 169iT^{2} \)
17 \( 1 + (-0.449 + 0.449i)T - 289iT^{2} \)
19 \( 1 + 9.89iT - 361T^{2} \)
23 \( 1 + (10.6 + 10.6i)T + 529iT^{2} \)
29 \( 1 - 36.2iT - 841T^{2} \)
31 \( 1 - 25.6T + 961T^{2} \)
37 \( 1 + (-13.3 + 13.3i)T - 1.36e3iT^{2} \)
41 \( 1 - 3.10T + 1.68e3T^{2} \)
43 \( 1 + (-2.72 - 2.72i)T + 1.84e3iT^{2} \)
47 \( 1 + (37.1 - 37.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (65.1 + 65.1i)T + 2.80e3iT^{2} \)
59 \( 1 - 80.3iT - 3.48e3T^{2} \)
61 \( 1 - 13.7T + 3.72e3T^{2} \)
67 \( 1 + (-84.3 + 84.3i)T - 4.48e3iT^{2} \)
71 \( 1 + 98.2T + 5.04e3T^{2} \)
73 \( 1 + (-52.4 - 52.4i)T + 5.32e3iT^{2} \)
79 \( 1 + 68.2iT - 6.24e3T^{2} \)
83 \( 1 + (-89.7 - 89.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 40.5iT - 7.92e3T^{2} \)
97 \( 1 + (-105. + 105. i)T - 9.40e3iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.024975844810148351456435429886, −8.483226951548077228107296496924, −7.54607199469190833265416257923, −6.61106695783776960435571379272, −6.18602089986295702647879889498, −4.85020118695578663955095984543, −4.20390811108780279928488964184, −3.33172862465264747248774571570, −1.90833542025334258861219376472, −1.01684206206432403194574639839, 0.802875464227899357194357987825, 1.90191784743471402843478930757, 3.17119823387628451746276965444, 3.95245496396433092722774810756, 5.04714981209664005778255518086, 5.94357818871456340610994108470, 6.40523666600902437045462839942, 7.78036160891436899135123233469, 8.175565454633090524205308396847, 8.933010280442623421715271936595

Graph of the $Z$-function along the critical line